Vector Basics

A vector is a quantity that has both direction (where it’s pointing) and magnitude (how large it is). A vector can be written in many formats: $\vec{v}=3i+2j-k=[3,2,-1]=\overrightarrow{AB}$

Each term in a vector refers to a component i.e. the $i$-component of $\vec{v}$ is 3.

Properties & Operations §

• Vectors can be added and subtracted by applying the operations to the respected components $[3,2,1]+[8,-1,5]=[11,1,6]=11i+j+6k$ Geometrically, $\vec{u}-\vec{v}$ is v in the opposite direction + u.
• Vectors cannot be divided or multiplied with each other. However, they can be multiplied or divided with a scalar
• The dot product and cross product are vector-specific operations
• The zero vector is represented as $\vec{0}=[0,0,0,\dots ]$
• Similar to geometric lines, vectors can be > Parallel & Perpendicular

Magnitude §

The magnitude of a vector refers to it’s geometric length. It uses the Pythagorean theorem. $|\vec{v}|=\sqrt{a^2+b^2+c^2+\dots }$

Parallel & Perpendicular §

Two vectors are parallel if there is some scalar $k$ that can be multiplied with one vector to obtain another.